Lemma 9.4.2.3. Let $\mathbb {K}$ be a collection of simplicial sets, let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets, and suppose we are given a commutative diagram
9.36
\begin{equation} \begin{gathered}\label{equation:fiberwise-cocompletion-one-direction} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [rr]^-{H} \ar [dr]_{U} & & \widehat{\operatorname{\mathcal{E}}} \ar [dl]^{ \widehat{U} } \\ & \operatorname{\mathcal{C}}& } \end{gathered} \end{equation}
which satisfies the conditions of Theorem 9.4.2.1. Then $H$ exhibits $\widehat{\operatorname{\mathcal{E}}}$ as a fiberwise $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{E}}$.
Proof.
Without loss of generality, we may assume that $\operatorname{\mathcal{C}}= \Delta ^1$. For $i \in \{ 0,1\} $, let $\operatorname{\mathcal{E}}_{i}$ denote the fiber $\{ i\} \times _{\Delta ^1} \operatorname{\mathcal{E}}$ and define $\widehat{\operatorname{\mathcal{E}}}_{i}$ similarly, so that $H$ restricts to a functor $H_{i}: \operatorname{\mathcal{E}}_{i} \rightarrow \widehat{\operatorname{\mathcal{E}}}_{i}$. By assumption, the functor $H_{i}$ exhibits $\widehat{\operatorname{\mathcal{E}}}_{i}$ as a $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{E}}_{i}$, and is therefore fully faithful (Proposition 8.4.5.3). Since $H$ carries $U$-cocartesian morphisms of $\operatorname{\mathcal{E}}$ to $\widehat{U}$-cocartesian morphisms of $\widehat{\operatorname{\mathcal{E}}}$, it follows that $H$ is fully faithful (Proposition 5.1.6.7). It follows from Lemma 9.4.2.2 that the cocartesian fibration $\widehat{U}$ is $\mathbb {K}$-cocomplete.
Fix an uncountable regular cardinal $\kappa $ such that $\widehat{\operatorname{\mathcal{E}}}$ is locally $\kappa $-small and every simplicial set $K \in \mathbb {K}$ is essentially $\kappa $-small. To complete the proof, it will suffice to show that for every object $X \in \operatorname{\mathcal{E}}_{0}$, the functor
\[ \mathscr {F}: \widehat{\operatorname{\mathcal{E}}}_{1} \rightarrow \operatorname{\mathcal{S}}^{< \kappa } \quad \quad Y \mapsto \operatorname{Hom}_{\widehat{\operatorname{\mathcal{E}}} }( H(X), Y ) \]
commutes with $K$-indexed colimits, for each $K \in \mathbb {K}$. Since $U$ is a cocartesian fibration, we can choose a $U$-cocartesian morphism $u: X \rightarrow X'$ of $\operatorname{\mathcal{E}}$ with $X' \in \operatorname{\mathcal{E}}_{1}$. By assumption, $H(u): H(X) \rightarrow H(X')$ is a $\widehat{U}$-cocartesian morphism in the $\infty $-category $\widehat{\operatorname{\mathcal{E}}}$, so the functor $\mathscr {F}$ is corepresented by the object $H(X')$. The desired result now follows from the characterization of $\mathbb {K}$-cocompletions given in Variant 8.4.6.9.
$\square$